# Noise suppression by quantum control before and after the noise

## TL;DR

This paper investigates the fundamental limits of quantum noise suppression using pre- and post-noise measurements and operations, revealing that optimal protocols in two-dimensional systems are classical and outlining conjectures for higher dimensions.

## Contribution

It proves that in two-dimensional systems, optimal noise suppression protocols are classical, and explores the limitations and potential of quantum control in noise mitigation.

## Key findings

- Noise suppression is nearly impossible with post-noise operations alone.
- Optimal protocols for depolarizing noise are classical, involving no quantum measurement.
- Conjecture and numerical evidence suggest similar results in higher-dimensional systems.

## Abstract

We discuss the possibility of protecting the state of a quantum system that goes through noise, by measurements/operations before and after the noise process. The aim is to seek for the optimal protocol that makes the input and output states as close as possible and clarify the role of the measurements therein. We consider two cases, one can perform quantum measurements/operations (i) only after the noise process and (ii) both before and after that. We prove in the two-dimensional Hilbert space that, in the case (i), the noise suppression is essentially impossible for all types of noise and, in the case (ii), the optimal protocol for the depolarizing noise is either the "do nothing" protocol or the "discriminate & reprepare" protocol. These protocols are not "truly quantum" and can be considered as classical. They involve no measurement or only use the measurement outcomes. These results describe the fundamental limitations in quantum mechanics from the viewpoint of control theory. Finally, we conjecture that a statement similar to the case (ii) holds for higher-dimensional Hilbert spaces and present some numerical evidence.